Question

Define mass and volume flow rates. How are they related to each other?

Short answer

Answer: Mass flow rate and volume flow rate are related through the fluid density, with the relationship being: $$\dot{m} = \rho \cdot Q.$$ Mass flow rate is equal to the density of the fluid multiplied by the volume flow rate.

Step-by-step solution

Definition of Mass Flow Rate

Mass flow rate is the measurement of the amount of mass that is passing through a given point per unit of time. It is usually represented by the symbol \(\dot{m}\) and is expressed in the units of mass per time, such as kg/s. Mathematically, mass flow rate can be expressed as $$\dot{m} = \rho \cdot A \cdot v,$$ where \(\rho\) is the fluid density (kg/m³), \(A\) is the cross-sectional area (m²) through which the fluid is passing, and \(v\) is the fluid velocity (m/s).

Definition of Volume Flow Rate

Volume flow rate is the measurement of the amount of volume (or space) that the fluid occupies as it passes through a given point per unit of time. It is usually represented by the symbol \(Q\) and is expressed in units of volume per time, such as m³/s. Mathematically, volume flow rate can be expressed as $$Q = A \cdot v,$$ where \(A\) is the cross-sectional area (m²) through which the fluid is passing, and \(v\) is the fluid velocity (m/s).

Relationship between Mass and Volume Flow Rates

Mass flow rate and volume flow rate are related through the density of the fluid. By substituting the expression for volume flow rate \(Q = A \cdot v\) into the expression for mass flow rate \(\dot{m} = \rho \cdot A \cdot v\), we obtain the relationship between mass flow rate and volume flow rate: $$\dot{m} = \rho \cdot Q.$$ Thus, mass flow rate is equal to the density of the fluid multiplied by the volume flow rate.